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The modularity of elliptic curves over all but finitely many totally real fields of degree 5. (English) Zbl 1519.11030

This paper studies the question of modularity of elliptic curves over totally real fields, in particular, totally real fields of degree \(5\). The main result is that every elliptic curve \(E\) over a totally real field \(F\) of degree \(5\) is modular, except for finitely many possibilities for \((F, j(E))\) where \(j(E)\) is the \(j\)-invariant of \(E\). The proof involves an interplay with modularity lifting theorems and rational points on modular curves, namely, the elliptic curves which cannot be treated by the modularity lifting theorems give rise to rational points of degree \(5\) on four modular curves of genus \(13, 21, 73, 153\).
A result of independent interest is a criterion to prove the finiteness of rational points of degree \(5\) on a curve with genus \(\ge 11\) with no morphism over of degree \(5\) to \(\mathbb{P}^1\) and such that the elliptic factors of its Jacobian have finite Mordell-Weil group. Its proof relies on a theorem of D. Abramovich [Int. Math. Res. Not. 1996, No. 20, 1005–1011 (1996; Zbl 0878.14019)] and G. Frey [Isr. J. Math. 85, No. 1–3, 79–83 (1994; Zbl 0808.14022)] relating finiteness of rational points of degree \(d\) on a curve \(C\) to the geometric gonality of \(C\) and the criterion itself is reminiscent of Mazur’s method but it is not effective due to the use of Faltings’ theorem on subvarieties of Jacobians.
The above criterion is used to prove the finiteness of rational points of degree \(5\) on the four modular curves in question using gonality estimates for modular curves and explicit computation of the isogeny factors of the Jacobians of these modular curves. The paper also points out interesting obstructions for generalizations to degree \(6\) totally real fields.

MSC:

11G05 Elliptic curves over global fields
11G18 Arithmetic aspects of modular and Shimura varieties

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